The Signed Root Deviance Profile and Confidence Intervals in Maximum Likelihood Analysis

Abstract

Abstract We investigate a natural extension of the profile t plot of Bates and Watts to a general parametric function g(θ) of the parameters θ in a general maximum likelihood analysis. Although the basic purpose of the extension, called the signed root deviance profile (SRDP), is to construct likelihood ratio (LR) confidence intervals for g(θ), it has various other applications that significantly extend its usefulness. The tangent to the plot of the SRDP at the maximum likelihood estimate ĝ of g(θ) gives the linear approximation (LA) interval based on the observed information matrix. The plot may be used as a diagnostic tool to compare LA and LR intervals and to suggest transformations of g(θ) whose LA intervals when inverted are close to the LR intervals for g(θ). The standard way to construct any profile is through repeated optimizations, but problems associated with nonlinear constraints can make this difficult. An alternative method based on integration is presented that avoids these problems. An exam...